Idea

Definition

Definition

For ℰ\mathcal{E} a topos, an internal site in ℰ\mathcal{E} is an internal categoryℂ=C1⇉C0\mathbb{C} = C_1 \rightrightarrows C_0 equipped with an internal coverage.

Spelled out in components, this means the following (as in (Johnstone), we shall only define sifted coverages). First, we define the subobjectSv(ℂ)↪PC1Sv(\mathbb{C}) \hookrightarrow PC_1 of sieves, where a subobject S↪C1S \hookrightarrow C_1 is a sieve if the composite

This result generalizes straightforwardly to an analogous statement for internal sheaves.

Definition

If 𝒞\mathcal{C} is equipped with a coverageJJ and 𝔻\mathbb{D} is equipped with an internal coverage KK , define a coverage J⋊KJ \rtimes K on 𝒞⋊𝔻\mathcal{C} \rtimes \mathbb{D} by declaring that a sieve on an object (U,V)(U,V) is (J×K)(J \times K)-covering if there exists an element S∈K(U)S \in K(U) with b(S)=Vb(S) = V, …